High Order Difference Schemes for Space Riesz Variable-Order Nonlinear Fractional Diffusion Equations

This article is aimed at studying new finite difference methods for one-dimensional (1D) and two-dimensional (2D) space Riesz variable-order (VO) nonlinear fractional diffusion equations (SRVONFDEs). In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Gr\"unwald-difference (4WSGD) operators for Riemann-Liouville constant-order (CO) fractional derivatives, which have a free parameter and have at least third order accuracy, we derive 4WSGD operators for space Riesz VO fractional derivatives. In order that the fully discrete schemes have good stability and can handle the nonlinear term efficiently, we apply the implicit Euler (IE) method to discretize the time derivative, which leads to IE-4WSGD schemes for SRVONFDEs. The stability and convergence of the IE-4WSGD schemes are analysed theoretically. In addiction, a parameter selection strategy is derived for 4WSGD schemes and banded preconditioners are put forward to accelerate the GMRES methods for solving the discretization linear systems. Numerical resutls demonstrate the effectiveness of the proposed schemes and preconditioners.